
Type 1, perturbed dispersive shock wave (PDSW):
This nematic DSW type resembles the standard KdV
DSW [1]}, [2]}, with a monotonic modulated wavetrain consisting of bright solitary waves at the leading
edge and linear waves at the trailing edge. There is no resonant wavetrain ahead of the DSW as there is
no resonance between the phase velocity of a possible resonant wavetrain, determined by (REF ),
with the DSW [3]}, [4]}. As the nematic equations (REF ) and () are
bi-directional, there is also a backwards propagating expansion wave, as for the compressible flow shock
tube problem [5]}, which links the level behind \(u_{-}\)  and an intermediate shelf of height \(|u| = u_{i}\) 
behind the DSW. The DSW takes the place of the shock in the shock tube problem. The solution for this DSW type
will be found as a perturbation of the KdV DSW.

Type 2, radiating dispersive shock wave (RDSW): When the jump height reaches a critical value [3]}, which
will be discussed in detail in Section , resonance can exist between the DSW and diffractive waves, so that a
resonant wavetrain ahead of the DSW is generated, as shown in Figure REF (b). The DSW itself resembles a KdV
DSW, but with an attached resonant wavetrain. This case resembles the equivalent RDSW type for the Kawahara
equation (REF ) [7]}. The DSW itself will be found to be a perturbed KdV DSW, as for the PDSW.

Type 3, crossover dispersive shock wave (CDSW): With increasing jump height, the DSW becomes unstable, as does
its resonant wavetrain, as seen in Figure REF (c). The DSW loses its rank ordered structure and the resonant
wavetrain has a highly modulated amplitude. Again, this DSW structure is similar to its equivalent for the Kawahara equation
[7]}.

Type 4, travelling dispersive shock wave (TDSW): At a critical jump height, the DSW itself disappears, leaving just
a resonant wavetrain, as shown in Figure REF (d). This solution form is similar to the travelling DSW (TDSW) found
for the Kawahara equation (REF ) [7]}. A remnant of the DSW is left in the form of a negative polarity
solitary wave which links the resonant wavetrain to the intermediate level. It is further seen that the resonant wavetrain
is of higher amplitude than in the CDSW case and has stabilised.

Type 5, vacuum dispersive shock wave (VDSW): As \(u_{+}\)  decreases, \(u_{-} - u_{+}\)  increases, and the amplitude of
the resonant wavetrain grows, the minimum of its oscillation eventually hits the vacuum point \(u=0\) , at which point there
is a phase singularity and the DSW solution changes form, as seen in Figure REF (e). A constant amplitude resonant
wavetrain now propagates on a varying mean level. Behind this, there is a resonant wavetrain on a constant mean level, which
is linked to the intermediate level by a negative polarity wave, as for the TDSW. A similar vacuum point solution occurs for
the defocusing NLS equation [10]}, although without the accompanying resonant wavetrain.

Type 6 (dam break solution): When the level ahead \(u_{+}=0\) , the solution becomes essentially non-diffractive, as seen
in Figure REF (f). This solution is referred to as the dam break solution as it is the simple wave solution of
the shallow water equations (REF ) and () on the characteristic \(C_{-}\)  and arises as a solution for the
flow generated by a breaking dam [5]}. As this dam break solution was considered in detail in previous work
[3]} and was found to be in near perfect agreement with numerical solutions of the nematic equations (REF )
and (), it will not be considered in detail here.

